EAAI Journal 2026 Journal Article
A self-explanatory deep learning-based soft sensor induced by a physical diffusion process and its application in an industrial process
- Xiao Wang
- Han Liu
- Xiaomei Qi
- Yong Zhang
Deep learning-based soft sensors (DLSSs) excel in predicting industrial process variables, yet their lack of explainability diminishes user trust and reliability. Drawing inspiration from classical physics, particularly heat conduction, we introduce an energy-constrained diffusion model. This model employs the diffusion process as a forward mechanism to elucidate data representation by neural networks. Through rigorous theoretical derivations, we establish an equivalence between the numerical updates of the diffusion process and deep data representations, leading to an optimal feature propagation solution. Subsequently, we propose a diffusion-based neural encoder (DUEncoder) and develop a deep prediction model, DUFormer, designed for self-explanatory DLSS. Practical experiments demonstrate the efficacy of DUFormer in predicting rotor thermal deformation in an air preheater. Utilizing information bottleneck theory, we visualize information paths to validate the self-explainability of the model. Comparative results demonstrate the superior performance of DUFormer against seven state-of-the-art baselines. It achieves the highest predictive accuracy, with a root mean square error (RMSE) of 0. 0804 and a coefficient of determination (R 2 ) of 0. 9834, while also reducing computational complexity by 60. 6% in floating-point operations (FLOPs) on average. This unique combination of high accuracy, inherent explainability, and low computational cost positions DUFormer as a highly practical and trustworthy solution for real-time industrial soft sensing.